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Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($$\alpha ,\beta ,k_1,k_2$$) linear boundary feedback law at the right end. The $$2 \times 2$$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $$\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$$. The role of the control parameters is examined and the following results have been proven: (i) when $$\beta \neq 0$$, the set of vibrational modes is asymptotically close to the vertical line on the complex $$\nu$$-plane given by the equation $$\Re \nu = \alpha + (1-k_1k_2)/\beta$$; (ii) when $$\beta = 0$$ and the parameter $$K = (1-k_1 k_2)/(k_1+k_2)$$ is such that $$\left |K\right |\neq 1$$ then the following relations are valid: $$\Re (\nu _n/n) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iii) when $$\beta =0$$, $|K| = 1$, and $$\alpha = 0$$, then the following relations are valid: $$\Re (\nu _n/n^2) = O\left (1\right )$$ and $$\Im (\nu _n/n) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iv) when $$\beta =0$$, $|K| = 1$, and $$\alpha>0$$, then the following relations are valid: $$\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$.
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Heterogeneous diffusion in an harmonic potential: the role of the interpretation
Abstract Diffusion in heterogeneous energy and diffusivity landscapes is widespread in biological systems. However, solving the Langevin equation in such environments introduces ambiguity due to the interpretation parameter $$\alpha$$, which depends on the underlying physics and can take values in the range $$0<\alpha<1$$. The typical interpretations are It\^o ($$\alpha=0$$), Stratonovich ($$\alpha=1/2$$), and Hanggi-Klimontovich ($$\alpha=1$$). Here, we analyse the motion of a particle in an harmonic potential---modelled as an Ornstein-Uhlenbeck process---with diffusivity that varies in space. Our focus is on two-phase systems with a discontinuity in environmental properties at $x=0$. We derive the probability density of the particle position for the process, and consider two paradigmatic situations. In the first one, the damping coefficient remains constant, and fluctuation-dissipation relations are not satisfied. In the second one, these relations are enforced, leading to a position-dependent damping coefficient. In both cases, we provide solutions as a function of the interpretation parameter $$\alpha$$, with particular attention to the It\^o, Stratonovich, and Hanggi-Klimontovich interpretations, revealing fundamentally different behaviours, in particular with respect to an interface located at the potential minimum.
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- Award ID(s):
- 2102832
- PAR ID:
- 10596722
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- New Journal of Physics
- ISSN:
- 1367-2630
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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